By Neil Dubin (auth.)
Stochastic techniques usually pose the trouble that, once a version devi ates from the best types of assumptions, the differential equations bought for the density and the producing services turn into mathematically ambitious. Worse nonetheless, one is especially usually resulted in equations that have no identified answer and do not yield to plain analytical equipment for differential equations. within the version thought of right here, one for tumor progress with an immunological re sponse from the traditional tissue, a nonlinear time period within the transition chance for the demise of a tumor mobilephone results in the above-mentioned problems. regardless of the mathematical hazards of this nonlinearity, we can ponder a extra refined version biologically. finally, so one can in attaining a extra sensible illustration of a sophisticated phenomenon, it is vital to ascertain mechanisms which permit the version to deviate from the extra mathematically tractable linear structure. to date, stochastic versions for tumor progress have virtually completely thought of linear transition probabilities.
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Extra info for A Stochastic Model for Immunological Feedback in Carcinogenesis: Analysis and Approximations
In either case c-l . (A-~) E t t=O c~ c =4 t=O lim i(A-~) . (C;l) = +00 c~ Figures 4 and 5 show plots of c l and Cz for the cumulant method. All the factors above, taken with the heavy algebra required to obtain the approximation, lead to the conclusion that some other approximation technique needs to be found. 5. 4. 5. 4. 5. 4. 0E-04 110 VALUES FIGURE Sa 44 STOCHASTIC VARIANCE FOR CUMULANT METHOD o~------------------------------------~ -20000 -40000 -60000 x gJ .... 5. 4. OE-07 200 VALUES FIGURE 5b 200 45 6.
Clearly the resulting stochastic process will be Gaussian, the normal distribution being the unique distribution with only first and second cumulants non-zero. The actual physical or mathematical justification for neglect of cumulants is not apparent, but Whittle refers to good results of Chandrasekhar (1955) in an application of the theory of turbulence. The cumulants can be expressed in terms of the moments as 38· ]Jl' (34) etc. When we neglect the third cumulant, we are assuming that Likewise there will be a corresponding assumption for each consecutive cumulant.
00 L at n=l N 2 n! 1) N (70) (~2 + + ~ + ~)lq(y,t). N2 N We now equate, in (70), terms in NO: (71) (2~ - l)~y(yq) + a H[(H)1) (A-)1) + ~l~ ay2 As per van Kampen (1961), we can transform this into an equation for which the solution is known. (72) seT) -log Let ~(l and note that ds/dT (73) an = -a ay -=. as 2~ ~) - log 4, - 1. Under this transformation, (71) becomes (qy) + Applying the additional transformations y = ze -s s ,q = e Q(z,s), we obtain (74) e 2s~ az 2 The solution to this equation is given by Chandrasekhar (1943): (75) where I 1 Q(z,s) = - - - exp [- (47f1)!